Bilbao Analysis and PDE
Welcome to the website of researchers working in Bilbao (either at UPV/EHU or at BCAM) in mathematical analysis, partial differential equations, inverse problems, mathematical physics, and/or related topics.
BBT Seminar
Online, Tuesday, May 7th, 2024, 11:00 - 12:00
Title: Minimality of the vortex solution for Ginzburg-Landau systems
Abstract
We consider the standard Ginzburg-Landau system for N-dimensional maps defined in the unit ball for some parameter eps>0. For a boundary data corresponding to a vortex of topological degree one, the aim is to prove the (radial) symmetry of the ground state of the system. We show this conjecture in any dimension N≥7 and for every eps>0, and then, we also prove it in dimension N=4,5,6 provided that the admissible maps are curl-free. This is part of joint works with L. Nguyen, M. Rus, V. Slastikov and A. Zarnescu.
Seminar
UPV/EHU, Thursday, May 9th, 2024, 12:00 - 13:00
Title: Catalan operators in Operator Theory
Abstract
Let $c=(C_n)_{n\ge 0}$ be the Catalan sequence and $T$ a linear and bounded operator on a Banach space $X$ such $4T$ is a power-bounded operator. The Catalan generating function is defined by the following Taylor series,
C(T) := \sum_{n=0}^\infty C_nT^n.
Note that the operator $C(T)$ is a solution of the quadratic equation $TY^2-Y+I=0.$ In this talk we study this algebraic equation in the case that $T$ is the infinitesimal generator of a C_0-semigroup. We express $C(T)$ by means of an integral representations which involves the resolvent operator $(\lambda-T)^{-1}$ or the C_0-semigroup. In the case that $T$ is a bounded operator, we define powers of the Catalan generating function $C(T)$ in terms of the Catalan triangle numbers. Finally, we give some particular examples to illustrate our results and some ideas to continue this research in the future. This is a research proyect with Alejandro Mahillo (Universidad de Zaragoza) and Natalia Romero (Universidad de La Rioja).
